Yes, for any "radius of convergence" problem, the best way to go is either the ratio test or the root test (and most of the time the ratio test is much simpler than the root test).

Here, \(\displaystyle a_n= \frac{1}{\sqrt{n!}}x^n\) so \(\displaystyle a_{(n+1)}= \frac{1}{\sqrt{n+1}}x^{n+1}\).

Then \(\displaystyle \left|\frac{a_{n}}{a_{n+1}}\right|= \frac{\sqrt{n!}}{\sqrt{(n+1)!}}|x|\)\(\displaystyle = \sqrt{\frac{n!}{(n+ 1)!}}|x|\)

What is the limit of that as n goes to infinity?

For what x is that limit less than 1?